On central $L$-values and the growth of the $3$-part of the Tate-Shafarevich group
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2021
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| _version_ | 1866911906569125888 |
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| author | Kezuka, Yukako |
| author_facet | Kezuka, Yukako |
| contents | Given any cube-free integer $λ>0$, we study the $3$-adic valuation of the algebraic part of the central $L$-value of the elliptic curve $$X^3+Y^3=λZ^3.$$ We give a lower bound in terms of the number of distinct prime factors of $λ$, which, in the case $3$ divides $λ$, also depends on the power of $3$ in $λ$. This extends an earlier result of the author in which it was assumed that $3$ is coprime to $λ$. We also study the $3$-part of the Tate-Shafarevich group for these curves and show that the lower bound is as expected from the conjecture of Birch and Swinnerton-Dyer, taking into account also the growth of the Tate-Shafarevich group. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2110_05521 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | On central $L$-values and the growth of the $3$-part of the Tate-Shafarevich group Kezuka, Yukako Number Theory 14H52, 11G05 Given any cube-free integer $λ>0$, we study the $3$-adic valuation of the algebraic part of the central $L$-value of the elliptic curve $$X^3+Y^3=λZ^3.$$ We give a lower bound in terms of the number of distinct prime factors of $λ$, which, in the case $3$ divides $λ$, also depends on the power of $3$ in $λ$. This extends an earlier result of the author in which it was assumed that $3$ is coprime to $λ$. We also study the $3$-part of the Tate-Shafarevich group for these curves and show that the lower bound is as expected from the conjecture of Birch and Swinnerton-Dyer, taking into account also the growth of the Tate-Shafarevich group. |
| title | On central $L$-values and the growth of the $3$-part of the Tate-Shafarevich group |
| topic | Number Theory 14H52, 11G05 |
| url | https://arxiv.org/abs/2110.05521 |